APMO 2017 P4

Discussion on Asian Pacific Mathematical Olympiad (APMO)
dshasan
Posts: 66
Joined: Fri Aug 14, 2015 6:32 pm
Location: Dhaka,Bangladesh

APMO 2017 P4

Unread post by dshasan » Sat May 27, 2017 1:40 pm

Call a rational number $r$ powerful if $r$ can be expressed in the form $\dfrac{p^k}{q}$ for some relatively prime positive integers $p,q$ and some integer $k > 1$. Let $a,b,c$ be positive rational numbers such that $abc = 1$. Suppose there exist positive integers $x,y,z$ such that $a^x + b^y + c^z$ is an integer. Prove that $a,b,c$ are all powerful.
The study of mathematics, like the Nile, begins in minuteness but ends in magnificence.

- Charles Caleb Colton

Post Reply